The View From 2009: This article was the one in a series where I tried to describe sound from the perspective of an alien creature (Zork-11) visiting earth. It was a lot of fun!

Hearing: The Highs and the Lows of It

In my last article, we talked about the Audio Window, that set of ranges of frequency, amplitude and time of air pressure variations that we regard as “sound.” This is all part of a larger, on-going consideration of how we hear, and how that knowledge can be used for fun and profit by us recording engineer and producer types.

This month I want to talk about how we perceive pitch, that simple sensation of relative highness or lowness of musical notes that turns out, when we examine it a little more closely, to be fairly weird.

Zork-11 Is Curious About This Thing We Call Pitch

When we talk about pitch, we are talking about musical notes, like Middle C, or A-440, or G-sharp two octaves below Middle C. Pitches have sensations of highness and lowness. Melodies are made up of sequences and patterns of higher and lower pitches. Chords are made up of stacks of pitches. We have twelve chromatic pitches to the octave. There are roughly four octaves worth of pitches, from Low C (ca. 65 Hz.) to High C (ca. 1040 Hz.). Pitches are the very stuff of music.

So, what are these so-called pitches? Zork-11, our imaginary friendly alien visitor from the fourth planet of that amazing giant red star, Betelgeuse, would like to know. Not having ears himself (he perceives everything as pulse trains of photons, hmmm . . .), he would really like to understand.

Our simple eighth-grade answer is that pitch is like frequency. You know, you’ve got your A-440, which is, like, 440 vibrations a second, only we call it A. Know what I mean?

Unfortunately, that doesn’t really cut it. You see, pitches aren’t really frequencies. They are subjective perceptions. They don’t exist in the physical realm, only in our brains. And we can hear an A-440 when there is NO frequency present anywhere near 440 Hz., the frequency we associate with that particular A.

There’s a related confusion. Musical notes (pitches) have timbre, which is derived from overtones, among other things. Now, overtones are stacks of frequencies, just like chords. In fact, the first seven overtones are a dominant 7th chord. How can we tell overtones from chords? What’s the difference? And how come the overtones don’t foul up our perception of chords? And what does this have to do with pitches? Zork-11 is fascinated, and VERY curious.

The Physics of Waveforms

In order to get at this particular thicket of auditory perception, we first need to consider physical waveforms. These are periodic cycles of air-pressure change. Last month we mentioned the simplest one, the sine wave, which has energy at only one frequency. Take a look at the following inventory of waveshapes, stolen from my book, Total Recording.

	Figure 1. Numerous audio waveforms, by permission of the author (that's me!)

Now these are shapes of changing pressure over time. If the period of any of these waveforms is 1/440th of a second, we will say that the waveform has a pitch of A-440. What gives the wave its “A-440”-ness is the length of the period, not the presence of energy at 440 Hz.

Each of these waveforms, except for the sine waves, has energy at multiple frequencies (this is what overtones are). Let’s look at the same set of waveforms viewed as energy distributed across the spectrum. For graphical ease, I’ve expressed them with a period of 4 ms. (250 Hz.).

	Figure 2. Spectrographs of the audio waveforms shown in Figure 1, also by permission of the author.

In each of these spectrographs (except for the 500 Hz. sine wave) there is plenty of energy at 250 Hz, which is an out-of-tune B-natural below Middle C. So, it makes sense to think that the pitch should be related to 250 Hz. We can imagine that these waveforms could all very well have the pitch of a 250 Hz. out-of-tune B.

But what about a B just below Middle C pitch that doesn’t have any 250 Hz. in it? Let’s take a look at some waveforms and spectrographs taken from a TurboSynth window. Here the overtones aren’t labeled according to frequency, just which overtone (by number) that it is. First, let’s look at our old friend, the sine wave, to get a feel for this. Here’s the TurboSynth sine wave:

	Figure 3. Sine wave drawn by TurboSynth.

And here is the expression of that same sine wave as a spectrograph:

	Figure 4. TurboSynth spectrograph of the sine wave in Figure 3.

Note that the only energy is at the first harmonic, called the “fundamental. For a 250 Hz. tone, this fundamental is at 250 Hz. Each harmonic above this will be 250 Hz. higher than the previous one. Now look at the following waveform and spectra. Here’s a complex wave:

	Figure 5. A complex wave drawn by TurboSynth.

Now let’s look at its spectrograph.

	Figure 6. TurboSynth spectrograph of the complex wave in Figure 5.

Notice that there is NO fundamental frequency present, just overtones 2, 3, 4, 5 and 6. So, there’s energy at 500, 750, 1000, 1250 and 1500 Hz., but none at 250.

What’s interesting about this (and what Zork-11 doesn’t understand) is that this also has a pitch of that out-of-tune B whose fundamental is 250 Hz. It sounds just as high (or low) as the 250 Hz. sine wave, just, er, brighter! How can this be? The nearest energy is at 500 Hz.!

Here’s another 250 Hz. waveform, where the lowest energy is at 1 kHz.:

	Figure 7. Yet another complex wave drawn by TurboSynth.

and here is its associated spectrograph:

	Figure 8. TurboSynth spectrograph of the complex wave in Figure 7.

This waveform also has a pitch of that same out-of-tune B-natural. What should be clear from the above examples is that pitch seems much more closely related to the length of the repeating period than to the energy present.

But, Zork-11 asks, how do you HEAR this? He can understand how higher frequencies would sound higher, but how can a group of higher frequencies sound just as high (or low) as a lower frequency? How does the ear do this?

The answer is, Zork-11, we don’t really know.

However, we’ve got some ideas.

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